forked from lijiext/lammps
52 lines
2.5 KiB
TeX
52 lines
2.5 KiB
TeX
\documentclass[12pt]{article}
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\begin{document}
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\begin{eqnarray*}
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E_{LJ} & = & 4\epsilon \left\{ \left[ \left( \frac{\sigma}{r} \right)^{\!12} -
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\left( \frac{\sigma}{r} \right)^{\!6} \right] +
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\left[ 6\left( \frac{\sigma}{r_c} \right)^{\!12} -
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3\left(\frac{\sigma}{r_c}\right)^{\!6}\right]\left(\frac{r}{r_c}\right)^{\!2}
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- 7\left( \frac{\sigma}{r_c} \right)^{\!12} +
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4\left( \frac{\sigma}{r_c} \right)^{\!6}\right\} \\
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E_{qq} & = & \frac{q_i q_j}{r}\left(1-\frac{r}{r_c}\right)^{\!2} \\
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E_{pq} & = & E_{ji} = -\frac{q}{r^3} \left[ 1 -
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3\left(\frac{r}{r_c}\right)^{\!2} +
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2\left(\frac{r}{r_c}\right)^{\!3}\right] (\vec{p}\bullet\vec{r}) \\
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E_{qp} & = & E_{ij} = \frac{q}{r^3} \left[ 1 -
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3\left(\frac{r}{r_c}\right)^{\!2} +
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2\left(\frac{r}{r_c}\right)^{\!3}\right] (\vec{p}\bullet\vec{r}) \\
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E_{pp} & = & \left[1-4\left(\frac{r}{r_c}\right)^{\!3} +
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3\left(\frac{r}{r_c}\right)^{\!4}\right]\left[\frac{1}{r^3}
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(\vec{p_i} \bullet \vec{p_j}) - \frac{3}{r^5}
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(\vec{p_i} \bullet \vec{r}) (\vec{p_j} \bullet \vec{r})\right] \\
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\end{eqnarray*}
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\begin{eqnarray*}
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F_{LJ} & = & \left\{\left[48\epsilon \left(\frac{\sigma}{r}\right)^{\!12} -
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24\epsilon \left(\frac{\sigma}{r}\right)^{\!6} \right]\frac{1}{r^2} -
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\left[48\epsilon \left(\frac{\sigma}{r_c}\right)^{\!12} - 24\epsilon
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\left(\frac{\sigma}{r_c}\right)^{\!6} \right]\frac{1}{r_c^2}\right\}\vec{r}\\
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F_{qq} & = & \frac{q_i q_j}{r}\left(\frac{1}{r^2} -
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\frac{1}{r_c^2}\right)\vec{r} \\
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F_{pq} &=& F_{ij } = -\frac{3q}{r^5} \left[ 1 -
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\left(\frac{r}{r_c}\right)^{\!2}\right](\vec{p}\bullet\vec{r})\vec{r} +
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\frac{q}{r^3}\left[1-3\left(\frac{r}{r_c}\right)^{\!2} +
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2\left(\frac{r}{r_c}\right)^{\!3}\right] \vec{p} \\
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F_{qp} &=& F_{ij} = \frac{3q}{r^5} \left[ 1 -
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\left(\frac{r}{r_c}\right)^{\!2}\right] (\vec{p}\bullet\vec{r})\vec{r} -
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\frac{q}{r^3}\left[1-3\left(\frac{r}{r_c}\right)^{\!2} +
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2\left(\frac{r}{r_c}\right)^{\!3}\right] \vec{p} \\
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F_{pp} & = &\frac{3}{r^5}\Bigg\{\left[1-\left(\frac{r}{r_c}\right)^{\!4}\right]
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\left[(\vec{p_i}\bullet\vec{p_j}) - \frac{3}{r^2} (\vec{p_i}\bullet\vec{r})
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(\vec{p_j} \bullet \vec{r})\right] \vec{r} + \\
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& & \left[1 -
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4\left(\frac{r}{r_c}\right)^{\!3}+3\left(\frac{r}{r_c}\right)^{\!4}\right]
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\left[ (\vec{p_j} \bullet \vec{r}) \vec{p_i} + (\vec{p_i} \bullet \vec{r})
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\vec{p_j} -\frac{2}{r^2} (\vec{p_i} \bullet \vec{r})
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(\vec{p_j} \bullet \vec{r})\vec{r}\right] \Bigg\} \\
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\end{eqnarray*}
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\end{document}
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